MEDB 5502, Module 14, review

Topics to be covered, 1 of 2

What you will learn
- 01 Linear regresion, analysis of variance
- 02 Linear regression with multiple independent variables
- 03 Analysis of covariance
- 04 Multi-factor analysis of variance
- 05 Dimension reduction
- 06 Logistic regression
- 07 Diagnostic tests

Topics to be covered, 2 of 2

What you will learn
- 08 Survival analysis
- 09 Meta-analysis
- 10 Dark side of data science
- 11 Hierarchical models
- 12 Longitudinal data
- 13 Bayesian statistics

Module 01, Review

Simple linear regression
One factor analysis of variance

Module 01, SPSS scatterplot

Scatterplot showing a weak positive relationship between acres and land price

Module 01, SPSS boxplot

Scatterplot number of baths versus building price

Module 01, SPSS calculation of R Square

Model summary showing a very low R Square value

Module 01, SPSS ANOVA table

ANOVA table showing an F ratio close to 1

Module 01, SPSS linear regression coefficients

Break #1

What you have learned
- 01 Linear regresion, analysis of variance
What’s coming next
- 02 Linear regression with multiple independent variables

Module 02, Linear regression with multiple independent variables

Analysis of variance table
- R-squared
- Partial F tests
Stepwise regression
Interpretation
Collinearity
Mediation

Module 02, Checking assumptions

Non-normality
- Q-Q plot of residuals
Lack of independence
- Assessed qualitatively
Unequal variances, Non-linearity
- Residual scatterplot

Module 02, SPSS dialog box for the general linear model

Module 02, SPSS computation of R-squared

10,548.480/15,079.017 = 0.70

Module 02, SPSS computation of change in R-squared

\(Partial\ R^2=0.700-0.693=0.007\)

The regression model with chest, abdomen, and hip accounts for 70% of the variation, but a model with just abdomen and hip does almost as well, accounting for 69.3% of the variation. The difference, 0.7% is the amount of additional variation accounted for when you add chest to a model that already included abdomen and hip.

That seems to contradict the earlier finding with the large negative test statistic and the small p-value. But actually, what it is saying is that although there is sufficient statistical evidence to conclude that chest circumference has a real impact, that impact is small. You will see this a lot, especially with datasets with very large sample sizes. You can often achieve statistical significance for an individual variable, but the practical impact can still be negligible.

Module 02, SPSS computation of partial F-test

Module 02, SPSS computation of full regression model

Module 02, SPSS computation of collinearity statistics

What is mediation?

“A situation when the relationship between a predictor variable and an outcome variable can be explained by their relationship to a third variable (the mediator)”
- Andy Field, Section 11.4

An informal assessment of mediation

There’s a relationship between age and percentage body fat. It is a bit easier to follow if you convert age in years to age in decades. The relationship between decades of age and body fat is that you add about 1.8% to your body fat every decade on average.

The question is, how does this happen? Does some of the muscle turn directly into fat, or do you add fat through an expanding girth?

When you look at a model with both decade of age and abdomen circumference, the effect of age is cut markedly. Instead of adding 1.8% body fat on average, you add about 0.7% body fat on average. The rest of the body fat comes from the “love handles” that you develop as you age.

So gaining some fat is inevitable, even if you can still fit into those size 32 jeans that you wore as a young man. But much of the gain in percentage body fat comes from moving from size 32 over time to size 42.

Module 02, SPSS Q-Q plot

Module 02, Scatterplot, 1 of 4

Module 02, Scatterplot, 2 of 4

Module 02, Scatterplot, 3 of 4

Module 02, Scatterplot, 4 of 4

Break #2

What you have learned
- 02 Linear regression with multiple independent variables
What’s coming next
- 03 Analysis of covariance

Module 03, Analysis of covariance

Confounding/covariate imbalance
Interpretation
Interactions

Module 03, Checking assumptions

Non-normality
- Q-Q plot of residuals
Lack of independence
- Assessed qualitatively
Unequal variances, Non-linearity
- Residual scatterplots

Module 03, SPSS calculation of unadjusted estimates

Module 03, SPSS calculation of adjusted estimates

Module 03, SPSS visualization, 1 of 2

Module 03, SPSS visualization, 2 of 2

Module 03, SPSS Q-Q plot

Module 03, SPSS scatterplot

Testing for an interaction

Break #3

What you have learned
- 03 Analysis of covariance
What’s coming next
- 04 Multi-factor analysis of variance

Module 04, Multi-factor analysis of variance

Tukey post hoc test
Interaction

Module 04, Checking assumptions

Non-normality
- Q-Q plot of residuals
Lack of independence
- Assessed qualitatively
Unequal variances
- Boxplots

Module 04, SPSS crosstabulation

Module 04, SPSS analysis of variance table

Module 04, SPSS removing irrelevant rows

Module 04, SPSS parameter estimates

Module 04, SPSS Tukey test

Module 04, SPSS Q-Q plot

Module 04, SPSS scatterplot

Box plots of exercise data

Mean values for the interaction

Analysis of variance table for interaction model

Parameter estimates for the interaction model

The intercept is the estimated average weight loss when all the indicator variables are equal to zero. That means the two reference categories: females for gender and reading for the exercise group. If you join a book club expect to gain more than three pounds.

The estimate for female=0 is the how much different the weight loss is in the reading group when you change gender from female to male. It’s a small change and not statistically significant.

The estimate for group=1 measures how much difference you see in the average weight loss in the jogging program compared to the reading program in the reference category (female). There is almost an 8 pound difference and this is statistically significant.

The estimate for group=2 measures how much difference you see in the average weight loss in the swimming program compared to the reading program in the reference category (female). There is an astounding 32 plus pound difference between swimming and reading, but again for the reference category of females.

The estimate for female=0*prog=1 shows how much different the benefit of jogging over reading is for men compared to women. Men did pretty well on the jogging program, losing 11 pounds instead of gaining more than 3 pounds. This is more impressive than for women who lost 4 pounds instead of gaining 3.

The estimate for female=0*prog=2 shows how much different the benefit of swimming is for men than for women. Both groups are better off swimming than reading, men losing 22 pounds versus a 3 pound gain. But the falls short of the benefit of swimming over reading for women a 29 pound loss versus a 3 pound gain.

Both of the parameters associated with the interaction are statistically significant.

Bottom line is that the benefits of jogging over reading and the benefits of swimming over reading are not the same for men and women.

Interaction plot, 1 of 2

An interaction plot draws a multiple lines connecting means. In this graph the top line shows the mean weight loss in the swimming program with the men’s mean on the left and the women’s mean on the right. The middle line shows the mean weight loss in the jogging program, again with the men’s mean on the left and the women’s on the right. The bottom line shows the mean weight loss (actually a weight gain!) in the reading program. Clearly the swimming program is the best, a bit better for women than men because the line slopes upward. The jogging program is second best, a bit worse for women than for men. The reading program is worst and the benefit, if you could call it that being about the same for men and women (the lines are flat).

It is a lack of parallelism that is the hallmark of an interaction.

Interaction plot, 2 of 2

When you can’t estimate an interaction

Special case, n=1
- Only one observation for categorical combination

Example, full moon study, 1 of 2

Example, full moon study, 2 of 2

Interaction between exercise program and hours spent exercising

Testing for interaction in analysis of covariance

Table with irrelevant rows removed

Parameter estimates

Intercept for prog=1, -8.997 + 2.216 = -6.781
Intercept for prog=2, 9.993 + 2.216 = 12.209
Intercept for prog=3, 2.216
Slope for prog=1, 10.409 + -2.956 = 7.453
Slope for prog=2, 9.83 + -2.956 = 6.874
Slope for prog=3, -2.956

The first three coefficients are a bit of an extrapolation because they represent patients who spend zero hours on the proposed intervention.

The intercept is the estimated average weight loss in the reference category (group=3 or reading program) when the number of hours devoted to the program is zero. In this case it is a weight gain of 2 pounds.

The coefficient for group=1 is how much better the weight loss is when you switch from the reading program to the jogging program, but still put in zero hours.

The coefficient for group=2 is how much better the weight loss is on average when you switch from the reading program to the jogging program, but still put in zero hours.

The coefficient for hours is how much better the weight loss is when you add an extra hour of effort and you are in the reading program. Actually, it is how much worse. Each extra hour of reading and you gain an extra three pounds on average.

The coefficient for hours*group=1 shows the change in slope when you switch from the reading program to the jogging program. Each hour invested is 10 pounds better (a loss of 7 pounds per hour invested instead of a gain of 3 pounds per hour invested) when you switch from the reading program to the jogging program.

The interpretation for hours*group=2 is similar It shows the change in slope when you switch from the reading program to the swimming program. Each hour invested is 10 pounds better (a loss of 7 pounds per hour invested instead of a gain of 3 pounds per hour invested) when you switch from the reading program to the swimming program.

Analysis of variance table

Table of means

Centered analysis

Weight loss at various conditions

hours = 2 (mean), effort = 30 (mean),
- \(\hat Y\) = 10.005
hours = 4 (mean+2), effort = 30 (mean),
- \(\hat y\) = 10.005 + 2.291*2 = 14.587
hours = 2 (mean), effort = 40 (mean+20)
- \(\hat Y\) = 10.005 + 0.707*20 = 24.145
hours = 4 (mean+2), effort = 40 (mean+20)
- \(\hat Y\) = 10.005 + 2.291*2 + 0.707*20 + 0.393*2*20 = 44.447

Line plots of means for unbalanced data

Table of means

Table of frequencies and column percentages

Break #4

What you have learned
- 04 Multi-factor analysis of variance
What’s coming next
- 05 Dimension reduction

Module 05, Dimension reduction

Principal components analysis
- Eigenvectors, Eigenvalues
Factor analysis
- Factor rotation

Correlation matrix, 1 of 3

Correlation matrix, 2 of 3

Correlation matrix, 3 of 3

Communalities

Eigenvalues

Scree plot

Component matrix

Boxplots of first four principal components

Scatterplot of first four principal components

R-squared using four principal components

R-squared using all 24 variables

Rotated factor pattern, 1 of 3

Rotated factor pattern, 2 of 3

Rotated factor pattern, 3 of 3

Break #5

What you have learned
- 05 Dimension reduction
What’s coming next
- 06 Logistic regression

Module 06, Precursors to logistic regression

Test of two proportions
Chi-square test of independence
Odds ratio versus relative risk

Module 06, Logistic regression

Linear on log odds scale
Assumptions
- Independence
- Linearity

Break #6

What you have learned
- 06 Logistic regression
What’s coming next
- 07 Diagnostic tests

Module 07, Diagnostic tests

Sensitivty, specificity
- SpPin, SnNout
Positive/negative predictive value
Likelihood ratio
ROC curve

Break #7

What you have learned
- 07 Diagnostic tests
What’s coming next
- 08 Survival analysis

Module 08, Basic survival analysis

Censoring
Kaplan-Meier curve
Log rank test

Module 08, Cox regression

Define hazard function
- Increasing/decreasing/constant hazard
- Hazard ratio
Assumptions
- Independence
- Non-informative censoring

Break #8

What you have learned
- 08 Survival analysis
What’s coming next
- 09 Meta-analysis

Module 09, Meta-analysis

Forest plot
Publication bias
- Funnel plot
Heterogeneity
- Cochran’s Q, I-squared

Break #9

What you have learned
- 09 Meta-analysis
What’s coming next
- 10 Dark side of data science

Module 10, Dark side of data science

Empiricism
Reification
Bias in data science

Break #10

What you have learned
- 10 Dark side of data science
What’s coming next
- 11 Hierarchical models

Module 11, Hierarchical models

Clustered data
Between and within cluster variation
- Intraclass correlation

Module 11, Checking assumptions

Independence
- Only between clusters
Normality
- Within clusters
- Between clusters

Break #11

What you have learned
- 11 Hierarchical models
What’s coming next
- 12 Longitudinal data

Module 12, Longitudinal data

Random intercepts model
Random slopes model

Module 12, Checking assumptions

Independence
- Only between subjects
Normality
- Residuals
- Random intercepts/slopes
Linearity
- Scatterplot of residuals

Break #12

What you have learned
- 12 Longitudinal data
What’s coming next
- 13 Bayesian statistics

Module 13, Bayesian statistics

Prior
- Flat or non-informative prior
Likelihood
Posterior

Summary, 1 of 2

What you have learned
- 01 Linear regresion, analysis of variance
- 02 Linear regression with multiple independent variables
- 03 Analysis of covariance
- 04 Multi-factor analysis of variance
- 05 Dimension reduction
- 06 Logistic regression
- 07 Diagnostic tests

Summary, 2 of 2

What you have learned
- 08 Survival analysis
- 09 Meta-analysis
- 10 Dark side of data science
- 11 Hierarchical models
- 12 Longitudinal data
- 13 Bayesian statistics